Improved bicharacteristic schemes for two-dimensional elastodynamic equations
Authors:
X. Lin and J. Ballmann
Journal:
Quart. Appl. Math. 53 (1995), 383-398
MSC:
Primary 73V15; Secondary 73D99, 73M25
DOI:
https://doi.org/10.1090/qam/1330658
MathSciNet review:
MR1330658
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Abstract: Several authors have used explicit numerical schemes of bicharacteristics to solve the system of hyperbolic partial differential equations describing the two-dimensional propagation of stress waves in elastic solids. However, their numerical approaches differ slightly, which results in different limits of the CFL number for stable solutions and in different defects of numerical accuracy near singular points like, e.g., a numerically caused cutting trace emanating from a crack tip. After reviewing the different approaches, some techniques are presented to set up stable explicit schemes with CFL number up to the limiting values 1 for the fastest mode. Finally, the schemes are applied to the crack problem of a shock loaded body, where the main reasons for the appearance of a cutting trace become apparent.
R. J. Clifton, A difference method for plane problems in dynamic elasticity, Quart. Appl. Math. 25, 9[ill]–116 (1967)
J. Ballmann, H. J. Raatschen, and M. Staat, High stress intensities in focussing zones of waves, Local Effects in the Analysis of Structures (P. Ladeveze, ed.), Elsevier, Amsterdam, 1985, pp. 235–252
J. Ballmann and M. Staat, Computation of impacts on elastic solids by methods of bicharacteristics, Computational Mechanics ’88, Theory and Applications (S. N. Atluri and G. Yagawa, eds.), Vol. 2, Springer-Verlag, New York, 1988, Chapter 60, pp. i1–i4
J. Ballmann and K.-S. Kim, Numerische Simulation mechanischer Wellen in geschichteten elastischen Körpern, Z. Angew. Math. Mech. 70, T204–T206 (1990)
K.-S. Kim, Spannungswellen an Grenzflächen in linearelastischen Scheiben, thesis for doctorate RWTH Aachen, VDI Verlag, Reihe 18, Nr. 91 (1991)
J. Bejda, Propagation of two-dimensional stress waves in an elastic/viscoplastic material, Proceedings of the 12th International Congress of Applied Mechanics, Stanford University, 121–134 (1968)
H. Fukuoka and H. Toda, High velocity impact of mild steel cylinder, Proceedings of IUTAM Symposium, Springer-Verlag, 397–402 (1978)
K. Fukatsu, K. Kawashima, and M. Oda, Dynamic elastic response of a short circular cylinder due to longitudinal impact, Trans. Japan. Soc. Mech. Engrg. (in Japanese) 50A, 869 (1984)
K. Liu, S. Tanimura, H. Igaki, and K. Kaizu, The dynamic behavior of an elastic circular tube due to longitudinal impact, JSME International Journal series I 32, 535–539 (1989)
K. Liu and T. Yokoyama, Dynamic behavior of elastic/viscoplastic bars of square cross section subjected to longitudinal impact, Trans. Japan. Soc. Mech. Engrg. (in Japanese) 58A, 109–116 (1992)
G. Ravichandran and R. J. Clifton, Dynamic fracture under plane wave loading, Internat. J. Fracture 40, 157–201 (1989)
G. Ravichandran, An analysis of dynamic crack initiation and propagating in elastic-viscoplastic solids, ICF-7 Advances in Fracture Research, Vol. 1 (K. Salama et al., eds.), Pergamon, 1989, pp. 819–826
X. Lin and J. Ballmann, Numerical method for elastic-plastic waves in cracked solids, part 1: antiplane shear problem, Archive of Applied Mechanics (Ingenieur-Archiv) 63, 261–282 (1993)
G. Strang, Accurate partial difference methods in nonlinear problems, Numer. Math. 13, 37–46 (1964)
P. D. Lax and B. Wendroff, Difference schemes for hyperbolic equations with high order of accuracy, Comm. Pure Appl. Math. 17, 381–398 (1964)
B. Eilon, D. Gottlieb, and G. Zwas, Numerical stabilizers and computing time for second-order accurate schemes, J. Comput. Phys. 9, 387–397 (1972)
X. Lin and J. Ballmann, Numerical method for elastic-plastic waves in cracked solids, part 2: plane strain problem, Archive of Applied Mechanics (Ingenieur-Archiv) 63, 283–295 (1993)
X. Lin and J. Ballmann, A finite difference method for elastic-plastic waves in solids, Numerical Methods in Engineering ’92 (Ch. Hirsch et al., eds.), Elsevier, Amsterdam, 1992, pp. 681–686
S. A. Thau and T. H. Lu, Transient stress intensity factors for a finite crack in an elastic solid caused by a dilatational wave, Internat. J. Solids and Structures 7, 731–750 (1971)
K. S. Kim, Dynamic propagation of a finite crack, Internat. J. Solids and Structures 15, 685–699 (1979)
X. Lin and J. Ballmann, Re-consideration of Chen’s problem by finite difference method, Engineering Fracture Mechanics 44, 735–739 (1993)
H. D. Lauermann, Ein Charakteristiken-Algorithmus zur Berechnung der instationären Transversalschwingungen von Hubschrauberrotorblättern oder anderer quasi-eindimensionaler elastischer Strukturen, thesis for doctorate RWTH Aachen (1990). See also: H. D. Lauermann, Instationäre Verformungen elastischer Rotorblätter, Z. Angew. Math. Mech. 70, T213–T215 (1990)
R. J. Clifton, A difference method for plane problems in dynamic elasticity, Quart. Appl. Math. 25, 9[ill]–116 (1967)
J. Ballmann, H. J. Raatschen, and M. Staat, High stress intensities in focussing zones of waves, Local Effects in the Analysis of Structures (P. Ladeveze, ed.), Elsevier, Amsterdam, 1985, pp. 235–252
J. Ballmann and M. Staat, Computation of impacts on elastic solids by methods of bicharacteristics, Computational Mechanics ’88, Theory and Applications (S. N. Atluri and G. Yagawa, eds.), Vol. 2, Springer-Verlag, New York, 1988, Chapter 60, pp. i1–i4
J. Ballmann and K.-S. Kim, Numerische Simulation mechanischer Wellen in geschichteten elastischen Körpern, Z. Angew. Math. Mech. 70, T204–T206 (1990)
K.-S. Kim, Spannungswellen an Grenzflächen in linearelastischen Scheiben, thesis for doctorate RWTH Aachen, VDI Verlag, Reihe 18, Nr. 91 (1991)
J. Bejda, Propagation of two-dimensional stress waves in an elastic/viscoplastic material, Proceedings of the 12th International Congress of Applied Mechanics, Stanford University, 121–134 (1968)
H. Fukuoka and H. Toda, High velocity impact of mild steel cylinder, Proceedings of IUTAM Symposium, Springer-Verlag, 397–402 (1978)
K. Fukatsu, K. Kawashima, and M. Oda, Dynamic elastic response of a short circular cylinder due to longitudinal impact, Trans. Japan. Soc. Mech. Engrg. (in Japanese) 50A, 869 (1984)
K. Liu, S. Tanimura, H. Igaki, and K. Kaizu, The dynamic behavior of an elastic circular tube due to longitudinal impact, JSME International Journal series I 32, 535–539 (1989)
K. Liu and T. Yokoyama, Dynamic behavior of elastic/viscoplastic bars of square cross section subjected to longitudinal impact, Trans. Japan. Soc. Mech. Engrg. (in Japanese) 58A, 109–116 (1992)
G. Ravichandran and R. J. Clifton, Dynamic fracture under plane wave loading, Internat. J. Fracture 40, 157–201 (1989)
G. Ravichandran, An analysis of dynamic crack initiation and propagating in elastic-viscoplastic solids, ICF-7 Advances in Fracture Research, Vol. 1 (K. Salama et al., eds.), Pergamon, 1989, pp. 819–826
X. Lin and J. Ballmann, Numerical method for elastic-plastic waves in cracked solids, part 1: antiplane shear problem, Archive of Applied Mechanics (Ingenieur-Archiv) 63, 261–282 (1993)
G. Strang, Accurate partial difference methods in nonlinear problems, Numer. Math. 13, 37–46 (1964)
P. D. Lax and B. Wendroff, Difference schemes for hyperbolic equations with high order of accuracy, Comm. Pure Appl. Math. 17, 381–398 (1964)
B. Eilon, D. Gottlieb, and G. Zwas, Numerical stabilizers and computing time for second-order accurate schemes, J. Comput. Phys. 9, 387–397 (1972)
X. Lin and J. Ballmann, Numerical method for elastic-plastic waves in cracked solids, part 2: plane strain problem, Archive of Applied Mechanics (Ingenieur-Archiv) 63, 283–295 (1993)
X. Lin and J. Ballmann, A finite difference method for elastic-plastic waves in solids, Numerical Methods in Engineering ’92 (Ch. Hirsch et al., eds.), Elsevier, Amsterdam, 1992, pp. 681–686
S. A. Thau and T. H. Lu, Transient stress intensity factors for a finite crack in an elastic solid caused by a dilatational wave, Internat. J. Solids and Structures 7, 731–750 (1971)
K. S. Kim, Dynamic propagation of a finite crack, Internat. J. Solids and Structures 15, 685–699 (1979)
X. Lin and J. Ballmann, Re-consideration of Chen’s problem by finite difference method, Engineering Fracture Mechanics 44, 735–739 (1993)
H. D. Lauermann, Ein Charakteristiken-Algorithmus zur Berechnung der instationären Transversalschwingungen von Hubschrauberrotorblättern oder anderer quasi-eindimensionaler elastischer Strukturen, thesis for doctorate RWTH Aachen (1990). See also: H. D. Lauermann, Instationäre Verformungen elastischer Rotorblätter, Z. Angew. Math. Mech. 70, T213–T215 (1990)
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Article copyright:
© Copyright 1995
American Mathematical Society
